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15 results for “matrix”

Formulae (15)

\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc

Determinant of a 2×2 matrix.

A\vec{v} = \lambda\vec{v}

Defines eigenvalues and eigenvectors of a matrix.

A^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

Inverse of a 2×2 matrix (ad−bc ≠ 0).

\text{tr}(A) = \sum_i a_{ii}

Sum of the diagonal entries of a square matrix.

(A+B)_{ij} = a_{ij} + b_{ij}

Matrices of equal size add entry by entry.

(cA)_{ij} = c\,a_{ij}

Multiply every entry of a matrix by a scalar.

AI = IA = A

The identity matrix leaves any matrix unchanged under multiplication.

(A^T)_{ij} = a_{ji}

Reflects a matrix across its main diagonal.

\det = a(ei-fh) - b(di-fg) + c(dh-eg)

Determinant of a 3×3 matrix by cofactor expansion.

\text{tr}(A) = \sum_i a_{ii}

Sum of the diagonal entries of a square matrix.

AI = IA = A

Multiplying by the identity leaves a matrix unchanged.

(AB)^T = B^T A^T

The transpose of a product reverses the order.

\text{rank}(A) + \text{nullity}(A) = n

Relates rank and null space dimension to columns.

\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}e&f\\g&h\end{bmatrix} = \begin{bmatrix}ae+bg&af+bh\\ce+dg&cf+dh\end{bmatrix}

Product of two 2×2 matrices.

\det(AB) = \det(A)\det(B)

The determinant of a product is the product of determinants.